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A156547 Decimal expansion of the central angle of a regular dodecahedron. 2
7, 2, 9, 7, 2, 7, 6, 5, 6, 2, 2, 6, 9, 6, 6, 3, 6, 3, 4, 5, 4, 7, 9, 6, 6, 5, 9, 8, 1, 3, 3, 2, 0, 6, 9, 5, 3, 9, 6, 5, 0, 5, 9, 1, 4, 0, 4, 7, 7, 1, 3, 6, 9, 0, 7, 0, 8, 9, 4, 9, 4, 9, 1, 4, 6, 1, 8, 1, 8, 8, 9, 9, 6, 6, 6, 7, 6, 7, 1, 3, 8, 7, 9, 5, 4, 8, 3, 4, 0, 7, 8, 1, 9, 4, 7, 3, 5, 0, 0, 2, 0, 8, 0, 9, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

If A and B are neighboring vertices of a regular dodecahedron having center O, then the central angle AOB is this number; the exact value is arccos((1/3)*sqrt(5)) = arcsin(2/3).

The (minimal) central angle of the other four regular polyhedra are as follows:

- tetrahedron: A156546,

- cube: A137914,

- octahedron: A019669,

- icosahedron: A105199.

LINKS

Table of n, a(n) for n=1..105.

FORMULA

The dodecahedron has 12 faces and 20 vertices. To find the central angle, we need any neighboring pair of vertices. Here are all 20 vertices:

- (d,d,d) where d is 1 or -1 (that's 8 vertices);

- (0, d*(t-1),d*t), where d is 1 or -1 and d = golden ratio = (1+sqrt(5))/2;

- (d*(t-1), d*t, 0); and ((d*t,0,d*(t-1)).

An example of a neighboring pair is (1,1,1) and (0,t,t-1).

Apply the usual formula for the cosine of the angle between two vectors.

EXAMPLE

arccos((1/3)*sqrt(5))=0.729727656226966..., or, in degrees,

41.810314895778598065857916730578259531014119535901347753...

MAPLE

evalf(arcsin(2/3)); #  Robert FERREOL, Sep 14 2019

MATHEMATICA

RealDigits[ArcCos[Sqrt[5]/3], 10, 120][[1]] (* Harvey P. Dale, Feb 23 2015 *)

PROG

(PARI) asin(2/3) \\ Charles R Greathouse IV, May 28 2013

CROSSREFS

Sequence in context: A021936 A277525 A154176 * A180872 A003673 A021141

Adjacent sequences:  A156544 A156545 A156546 * A156548 A156549 A156550

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Feb 09 2009

STATUS

approved

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Last modified October 18 12:18 EDT 2019. Contains 328160 sequences. (Running on oeis4.)